A First Course in FUZZY and NEURAL CONTROL

3.8. FUZZY RELATIONS 119

Example 3.13LetX={x 1 ,x 2 ,x 3 },

R=

x 1 x 2 x 3

x 1

x 2

x 3





0. 90. 20. 2

0. 90. 40. 5

1. 00. 61. 0



 ,Q=

x 1 x 2 x 3

x 1

x 2

x 3





0. 30. 80

00. 61. 0

0. 30. 80. 2





thenR◦Qhasijentry

W 3

k=1(R(xi,xk)∧Q(xk,xj))so that

R◦Q=

x 1 x 2 x 3

x 1

x 2

x 3





0. 30. 80. 2

0. 30. 80. 4

0. 30. 80. 6





Notice the similarity to matrix product. Use essentially the same algorithm,
but replace product by minimum and sum by maximum.

WhenS⊆X◊Yis a relation, there are two natural maps called projections:

πX : S→X:(x, y) 7 →x

πY : S→Y:(x,y) 7 →y

In the case of a fuzzy relationR:X◊Y→[0,1],theprojectionsofRonX
andYare the fuzzy subsetsπX(R)ofXandπY(R)ofY defined by

πX(R)(x)=

W

{R(x,y)|y∈Y}

πY(R)(y)=

W

{R(x,y)|x∈X}

3.8.1 Equivalencerelations.....................

The fundamental similarity relation is anequivalencerelation. Withanordinary
set, an equivalence relationpartitions the set into separate pieces, any two
members of one of these pieces beingequivalent.Theformaldefinition is the
following.

Definition 3.14 Anequivalence relationis a subsetS⊆X◊X,suchthat
for allx,y,z∈X

1.(x,x)∈S(Sis reflexive.)

2.(x,y)∈Sif and only if(y,x)∈S(Sis symmetric.)

3.(x,y)∈Sand(y,z)∈Simplies(x,z)∈S(Sis transitive.)

This has been generalized to fuzzy sets as follows.

Definition 3.15 Afuzzy equivalence relationis a fuzzy relationSon a set
X, such that for allx,y,z∈X

1.S(x,x)=1(Sis reflexive.)

2.S(x,y)=S(y,x)(Sis symmetric.)

3.S(x,z)≥S(x,y)∧S(y,z)(Sis max-min transitive.)